Geometric realization of algebraic conformal field theories

TL;DR

This paper establishes a geometric method to derive conformal nets from vertex operator algebras in two-dimensional conformal field theory, bridging algebraic and geometric frameworks.

Contribution

It introduces a continuous geometric interpolation procedure to construct conformal nets from vertex operator algebras using mollified fields.

Findings

Conformal nets can be obtained from vertex operator algebras via geometric mollification.

The mollified fields produce bounded, local operators on the Hilbert space.

The approach connects algebraic and geometric formulations of conformal field theory.

Abstract

We explore new connections between the fields and local observables in two dimensional chiral conformal field theory. We show that in a broad class of examples, the von Neumann algebras of local observables (a conformal net) can be obtained from the fields (a unitary vertex operator algebra) via a continuous geometric interpolation procedure involving Graeme Segal's functorial definition of conformal field theory. In particular, we construct conformal nets from these unitary vertex operator algebras by showing that 'geometrically mollified' versions of the fields yield bounded, local operators on the Hilbert space completion of the vertex algebra. This work is inspired by Henriques' picture of conformal nets arising from degenerate Riemann surfaces.